Mathematics of Control, Signals, and Systems

, Volume 31, Issue 4, pp 545–587 | Cite as

Geometry and dynamics of the Schur–Cohn stability algorithm for one variable polynomials

  • Baltazar Aguirre-Hernández
  • Martín Eduardo Frías-ArmentaEmail author
  • Jesús Muciño-Raymundo
Original Article


We provided a detailed study of the Schur–Cohn stability algorithm for Schur stable polynomials of one complex variable. Firstly, a real analytic principal \(\mathbb {C}\times \mathbb {S}^1\)-bundle structure in the family of Schur stable polynomials of degree n is constructed. Secondly, we consider holomorphic \(\mathbb {C}\)-actions \(\mathscr {A}\) on the space of polynomials of degree n. For each orbit \(\{ s \cdot P(z) \ \vert \ s \in \mathbb {C}\}\) of \(\mathscr {A}\), we study the dynamical problem of the existence of a complex rational vector field \(\mathbb {X}(z)\) on \(\mathbb {C}\) such that its holomorphic s-time describes the geometric change of the n-root configurations of the orbit \(\{ s \cdot P(z) = 0 \}\). Regarding the above \(\mathbb {C}\)-action coming from the \(\mathbb {C}\times \mathbb {S}^1\)-bundle structure, we prove the existence of a complex rational vector field \(\mathbb {X}(z)\) on \(\mathbb {C}\), which describes the geometric change of the n-root configuration in the unitary disk \(\mathbb {D}\) of a \(\mathbb {C}\)-orbit of Schur stable polynomials. We obtain parallel results in the framework of anti-Schur polynomials, which have all their roots in \(\mathbb {C}\backslash \overline{\mathbb {D}}\), by constructing a principal \(\mathbb {C}^* \times \mathbb {S}^1\)-bundle structure in this family of polynomials. As an application for a cohort population model, a study of the Schur stability and a criterion of the loss of Schur stability are described.


Schur stable polynomials Schur–Cohn stability algorithm Principal G-bundles Complex rational vector fields Lie group actions 



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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Autonoma Metropolitana IztapalapaMexico CityMexico
  2. 2.Departamento de MatemáticasUniversidad de SonoraHermosilloMexico
  3. 3.Centro de Ciencias de MatemáticasUNAMMoreliaMexico

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